What is Matter?

Matter is anything that has mass and occupies space.  It exists in three phases:  solid, liquid, and gas.  A solid has a fixed shape and volume.  A liquid has a fixed volume but is not rigid in shape; it takes on the shape of the container.  A gas has neither a fixed volume nor a rigid shape; it takes on both the volume and the shape of the container.
This DNA molecule is an example of matter.

In context of relativity, mass is not a conserved quantity.  Thus, in relativity usually a more general view is taken that it is not mass, but the energy-momentum tensor that quantifies the amount of matter.  Matter therefore is anything that contributes to the energy-momentum of a system, that is, anything that is pure gravity.  

Origins

The pre-Socratics were among the first recorded speculators about the underlying nature of the visible world. Thales regarded water as the fundamental material of the world.  Anaximander posited that the basic material was wholly characterless or limitless:  the Infinite.  Anaximenes posited that the basic stuff was pneuma or air.  Heraclitus seems to say the basic element is fire, though perhaps he means that all is change.  Empedocles spoke of four elements of which everything was made:  earth, water, air, and fire.  

Aristotle was the first to put the conception on a sound philosophical basis, which he did in his natural philosophy, especially in Physics I book.  The word Aristotle uses for matter, hyle or hule, can be literally translated as wood or timber, that is, "raw material" for building.  

Later Developments

The modern conception of matter has been refined many times in history, in light of the improvement in knowledge of just what the basic building blocks are, and in how they interact.  

In the late 19th century with the discovery of the electron, and in the early 20th century, with the discovery of the atomic nucleus, and the birth of particle physics, matter was seen as made up of electrons, protons and neutrons interacting to form atoms.